Spectral properties of the Laplacian of temporal networks following a constant block Jacobi model

We study the behavior of the eigenvectors associated with the smallest eigenvalues of the Laplacian matrix of temporal networks. We consider the multilayer representation of temporal networks, i.e. a set of networks linked through ordinal interconnected layers. We analyze the Laplacian matrix, known as supra-Laplacian, constructed through the supra-adjacency matrix associated with the multilayer formulation of temporal networks, using a constant block Jacobi model which has closed-form solution. To do this, we assume that the inter-layer weights are perturbations of the Kronecker sum of the separate adjacency matrices forming the temporal network. Thus we investigate the properties of the eigenvectors associated with the smallest eigenvalues (close to zero) of the supra-Laplacian matrix. Using arguments of perturbation theory, we show that these eigenvectors can be approximated by linear combinations of the zero eigenvectors of the individual time layers. This finding is crucial in reconsidering and generalizing the role of the Fielder vector in supra-Laplacian matrices.